Does this diagram represent several LSTMs, or one through several timesteps?
are the LSTM cells on a given row all the same cell, with time flowing forward from left to right?

Yes this is correct
The x-axis on this figure is basically the time axis. Essentially all pink boxes in the same row are the same LSTM cell, with different inputs from the same sequence. At each timestep, the cell takes an input and produces an output which is fed to the next layer. At the 8-th layer, the outputs over all timesteps are inputted at the same time to the attention layer.